AC Susceptibility and Heat Capacity Studies of the Geometrically Frustrated Pyrochlores Terbium Titanium Tin Oxides and Holmium Titanates

UNLV Theses, Dissertations, Professional Papers, and Capstones

2009

AC susceptibility and heat capacity studies of the geometrically frustrated pyrochlores terbium titanium tin oxides and holmium titanates

Daniel Antonio University of Nevada Las Vegas

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Repository Citation Antonio, Daniel, "AC susceptibility and heat capacity studies of the geometrically frustrated pyrochlores terbium titanium tin oxides and holmium titanates" (2009). UNLV Theses, Dissertations, Professional Papers, and Capstones. 131. http://dx.doi.org/10.34917/1385237

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GEOMETRICALLY FRUSTRATED PYROCHLORES

Tb2Ti2−xSnxO7 AND Ho2−xRxTi2O7

Daniel Antonio

Bachelor of Science University of Nevada Las Vegas 2004

A thesis submitted in partial fulﬁllment of the requirements for the

Master of Science Degree in Physics Department of Physics and Astronomy College of Sciences

THE GRADUATE COLLEGE

We recommend that the thesis prepared under our supervision by

Daniel Antonio

entitled

AC Susceptibility and Heat Capacity Studies of the Geometrically Frustrated Pyrocholores Tb2Ti2-x SnxO7 and Ho2-xRxTi2O7

be accepted in partial fulfillment of the requirements for the degree of

Master of Science Physics

Andrew Cornelius, Committee Chair

Michael Pravica, Committee Member

Tao Pang, Committee Member

Adam Simon, Graduate Faculty Representative

Ronald Smith, Ph. D., Vice President for Research and Graduate Studies and Dean of the Graduate College

December 2009

ABSTRACT

AC Susceptibility And Heat Capacity Studies Of The Geometrically Frustrated Pyrochlores Tb2Ti2−xSnxO7 And Ho2−xRxTi2O7

Daniel Antonio

Dr. Andrew Cornelius, Defense Committee Chair Associate Professor of Physics University of Nevada, Las Vegas

Materials with a geometrically frustrated magnetic pyrochlore lattice have been of interest due to their unusual ground states. Tb2Ti2O7 is known for having a spin

liquid ground state which does not transition to a long-range ordered state down to

at least 50 mK. Ho2Ti2O7 has a macroscopically degenerate spin ice ground state which resembles that of the proton ordering in water ice. Heat Capacity measurements of Tb2Ti2−xSnxO7 were done down to 0.36 K and AC magnetic susceptibility measurements of Ho2−xYxTi2O7 and Ho2−xLaxTi2O7 were done for frequencies from

10 Hz to 10 kHz down to 1.8 K, both in magnetic fields up to 9 T. These experiments were performed in an effort to further understanding of the factors that lead to their unusual behavior and to study the effects of introducing disorder through doping with nonmagnetic elements. Determination of the effect of an external field on the hyperfine crystal field at the Tb sites was done. In addition, unusual behavior in the ac susceptibility of the Ho samples at lower temperatures was observed.

iii TABLE OF CONTENTS

ABSTRACT ...... iii

LISTOFFIGURES...... v

LISTOFTABLES...... vi

ACKNOWLEDGMENTS...... vii

CHAPTER1 INTRODUCTION ...... 1 GeometricFrustration ...... 1 MagneticOrder...... 2 SpinInteractions...... 5 Structure of Tb2Ti2O7 and Ho2Ti2O7 ...... 6

CHAPTER 2 EXPERIMENTAL PROCEDURE ...... 9 Heat Capacity Measurement...... 9 MagneticMesurements ...... 11

CHAPTER 3 RESULTS AND ANALYSIS ...... 13 Sn substituion in Tb2Ti2O7 ...... 13 La and Y substitution in Ho2Ti2O7 ...... 15

CHAPTER4 CONCLUSIONS ...... 18

REFERENCES ...... 48

VITA ...... 50

iv LIST OF FIGURES

Figure1 Frustrated triangle...... 20 Figure 2 Inverse χ forafrustratedsystem...... 20 Figure3 Pyrochlore lattice...... 21 Figure4 Spinice ...... 21 Figure5 PPMS ...... 22 Figure6 Thehelium3insert ...... 23 Figure 7 A Helium 3 Heat Capacity Puck...... 24 Figure 8 The drive and detection coils for the PPMS ACMS Option...... 25 Figure 9 The magnetic heat capacity of Tb2Ti2O7...... 26 Figure 10 The magnetic heat capacity of Tb2Ti1.95Sn0.05O7...... 27 Figure 11 The magnetic heat capacity of Tb2Ti1.9Sn0.1O7...... 28 Figure 12 The magnetic heat capacity of Tb2TiSnO7...... 29 Figure 13 The magnetic heat capacity of Tb2Sn2O7...... 30 Figure 14 The hyperfine field for Tb2Ti2O7...... 31 Figure 15 The hyperfine field for Tb2Ti1.95Sn0.05O7...... 32 Figure 16 The hyperfine field for Tb2Ti1.9Sn0.1O7...... 33 Figure 17 The hyperfine field for Tb2TiSnO7...... 34 Figure 18 The hyperfine field for Tb2Sn2O7...... 35 Figure 19 The expected response due to thermally activated behavior...... 36 Figure 20 AC susceptibility versus temperature for Dy2Ti2O7...... 37 Figure 21 The AC susceptibility of Ho0.2Y1.8Ti2O7...... 38 Figure 22 The AC susceptibility of Ho0.5Y1.5Ti2O7...... 39 Figure 23 The AC susceptibility of Ho0.7Y1.3Ti2O7...... 40 Figure 24 The AC susceptibility of Ho0.3La1.7Ti2O7...... 41 Figure 25 The AC susceptibility of Ho1.6La0.4Ti2O7...... 42 Figure 26 The AC susceptibility of Ho1.9La0.1Ti2O7...... 43 Figure 27 Graphs of the applied frequency versus the inverse peak frequency . . 44

v LIST OF TABLES

Table 1 Frustration parameters for some strongly frustrated materials...... 45 Table 2 The results of the ﬁts of the low temperature heat capacity...... 46 Table 3 The results of the ﬁts of the low temperature heat capacity cont. . . . 47

vi ACKNOWLEDGMENTS I would like to thank the following people: Dr. Andrew Cornelius for allowing me to work in his lab and being very patient. Dr. Micheal Pravica, Dr. Tao Pang, and Dr. Adam Simon for being on my

committee. Jason Gardner for providing my samples. Matthew Jacobsen for his help in preparing this thesis. Brittany Webb and Heidi Brooks for their assistance taking this data.

Work at UNLV is supported by DOE EPSCoR-State/National Laboratory Part- nership Award DE-FG02-00ER45835 and DOE Cooperative Agreement DE-FC08- 98NV1341.

vii CHAPTER 1

INTRODUCTION

Geometric Frustration

Most magnetic systems spontaneously transition to a state of long range magnetic order at some critical temperature. The type of ordering (ferromagnetic, antiferromagnetic, ferrimagnetic, etc.) depends on the type of interactions between the spins. Systems with competing magnetic interactions may resist this ordering. A material that exhibits these competing interactions is said to be frustrated. Frustration in systems is common, but cases where the competing interactions are of the same magnitude make the frustration very strong, which can lead to unusual and complex phenomena such as highly degenerate ground states.

Geometric frustration arises when incompatibilities between the spin interactions and the symmetry of the lattice itself do not allow all neighboring spins to satisfy the dominant magnetic interaction. The simplest case of this is illustrated by a triangle of antiferromagneticly coupled Ising spins, shown in Figure 1, ﬁrst studied by G.H.

Wannier in 1950 [1]. The spins can only point up or down, and will preferentially orient anti-parallel to each other. Two of the spins can easily orient in opposite directions and satisfy the interaction, but the last spin cannot. Both orientations conflict equally with one of the first two spins, so there is no preferred lowest energy orientation for the third spin. This causes a highly degenerate ground state. This frustration can sometimes be relieved by making one orientation more favorable. Placing the frustrated material in an external magnetic field causes the spins pointing in same direction of the field to have lower energy than those pointing against the field, by:

∆E = B · µ − (−B · µ)=2(B · µ) (1.1)

1 where B is the external field and µ is the magnetic moment of a single spin. Also, because geometric frustration depends on lattice symmetries, one would expect per- turbations to the lattice to relieve the frustration. Non hydrostatic pressure would cause unequal distances between competing pairs of spins and therefore affect the relative strengths of the rival interactions. Hydrostatic pressure will shorten the av- erage distances between spins and can possibly change the nature of the interactions. Randomly doping the material with atoms of slightly different size or replacing the magnetic atoms with non magnetic ones would also be expected to break the lattice symmetries.

As geometrically frustrated magnets are ideal models for complex phenomena ranging from protein folding to nuclear fragmentation to ﬁnancial systems, this work allows a direct experimental probe of extremely complex models that are applicable to a broad range of disciplines. The experimental tools allow for precise adjustment of the magnetic interactions and allow rigorous experimental testing of theory in these model systems.

Magnetic Order

Above a critical temperature, magnetic systems with either ferromagnetic or antiferromagnetic interactions will be in a paramagnetic state. There is no long range order because thermal ﬂuctuations overpower the magnetic interactions, leading to randomly oriented spins. Below a certain temperature, called the Curie temperature,

TC , for ferromagnetic systems and the Néel temperature, TN , for antiferromagnetic systems, the thermal fluctuations will no longer be strong enough to disrupt the magnetic interactions and the system will usually spontaneously transition to an ordered state. In a ferromagnet, the neighboring spins favor orienting parallel to each other, so the bulk material has a net magnetic moment. Using the mean field approximation,

2 in which it is assumed that each magnetic atom experiences a ﬁeld proportional to it the material’s magnetization, the temperature at which the system orders can be

estimated from from the Curie-Weiss law at temperatures above TC :

C χ = (1.2) (T − TC )

where χ = M/H is the magnetic susceptibility of the paramagnetic phase, TC is the critical temperature, C is the Currie constant:

Np2µ2 C = B (1.3) 3kBT

and p is the eﬀective number of Bohr magnetons:

1 p = g[J(J + 1)] 2 (1.4)

where N is the total number of atoms, J is the angular momentum quantum number, and g is the Land´efactor:

j(j +1)+ s(s + 1) − l(l + 1) g =1+ (1.5) 2j(j + 1)

This only works for temperatures well above TC , and it can be seen from a graph of the inverse susceptibility versus temperature that it diverges from a straight line as

it approaches TC . The exchange integral, Jex, which gives a measure of the strength

of the exchange interactions, can usually be approximated from TC using:

3kBT J = C (1.6) ex 2zS(S + 1)

where z is the number of nearest neighbors, and S is the magnetic spin. This assumes that farther neighbors and dipole interactions are negligible [9].

3 In an antiferromagnet, the spins favor orienting antiparallel to each other. This can be viewed as two interpenetrating ferromagnetic lattices of the same ion that have a strong antiferromagnetic interaction between them. This gives an equation analogous to the Curie-Weiss Law for antiferromagnets:

C χ = (1.7) T + TN

Note that this formula is very similar to equation (1.2) for ferromagnets. Typically, one combines equations (1.2) and (1.7) and ﬁts experimental data to

C χ = (1.8) T − θ

where values for the Weiss Temperature θ can be positive (ferromagnetic) or negative (antiferromagnetic), depending on the type of magnetic interaction.

Typically, values of TN and TC are on the order of |θ|. In systems that display frustration, the magnetic ordering temperature can be much lower than that of |θ|. Deﬁning a frustration parameter f as the ratio of |θ| to the ordering temperature

(Tmag = TN or TC ), one can get an idea of the degree of frustration. A material having f > 1 corresponds to frustration. A value of f > 10 cannot be described

by mean-ﬁeld theory, indicating a more complicated state. This can be considered a benchmark for strong geometric frustration [10]. Table 1 gives values of f for some materials that are strongly frustrated and some materials that would be considered “normal” for comparison.

Dipole and exchange effects from farther neighbors often account for this difference. Another odd effect of frustration is that for strongly frustrated magnetic systems, the susceptibility follows ideal linear Curie-Weiss behavior much more closely for temperatures well below the expected Néel temperature as shown in Figure 2. Below the ordering temperature for both cases, the magnetization diverges from

4 this simple behavior. The magnetization goes to inﬁnity for the Curie-Weiss law at TC , and more complex behavior takes over for lower temperatures. For antiferromagnets,

the magnetization has its maximum value at TN , where there is a well deﬁned cusp. This feature is matched by a peak in the heat capacity, corresponding to a dramatic

change in magnetic entropy that marks the transition as well.

Spin Interactions

The interactions between spins on diﬀerent lattice sites are what give rise to spon-

taneous magnetic order. The purely quantum-mechanical exchange ﬁeld is often what dominates these interactions. This eﬀect arises from the Pauli exclusion principal, which prevents identical fermions from being in the same state. The wavefunctions of the neighboring spins overlap and must be anti-symmetric under an exchange of coor- dinates. This keeps parallel spins from being near each other, because the probability

of ﬁnding them goes to zero as they approach the same position, and therefore the Coulomb repulsion is lessened. There is then a diﬀerence in energy between parallel

and antiparallel spins. This can be represented by an energy term −2Jµ1 · µ2, where J is the exchange integral, given by:

∗ ∗ 3 3 Ψ1(1)Ψ2(2)Ψ1(2)Ψ1 J = d r1d r2 (1.9) Z r12

In a solid, the Coulomb interactions are more complex due to repulsion between electrons and attraction to neighboring nuclei, so J can be positive, indicating ferromagnetic interactions that favor parallel alignment, or J can be negative, indicating antiferromagnetic interactions that favor antiparallel alignment. The exchange energy for a solid can be represented by the Heisenberg model:

ex H = − JijSi · Sj (1.10) Xi Xi6=j

5 where Si and Sj are the total spins of the atoms at sites i and j [13, 14]. The exchange interaction depends on the overlap of wavefunctions, so the interactions beyond the nearest neighbor are often negligible. The spins can also be treated as classical dipoles, which interact with each other

and can create another preferred spin alignment. The dipole-dipole interactions are proportional to 1/r3, so they are longer range than the exchange interaction. In a solid, atoms farther than nearest neighbor must be taken into account. This can make calculations complicated and whether the exchange or dipole interactions dominate depends on the system. The dipole-dipole interaction energy is given by:

1 S · S (S · r )(S · r ) Hdip = (gµ )2 i j − 3 i ij j ij (1.11) 2 B r3 r5 Xi,j ij ij

Structure of Tb2Ti2O7 and Ho2Ti2O7

In the compounds terbium titanate and holmium titanate, the rare earth ions

form a magnetic pyrochlore lattice, shown in Figure 3. This lattice of corner sharing tetrahedrons, essentially a three dimensional triangular lattice, is highly frustrated.

3+ Both of these compounds show unusual ground states. The Tb ions in Tb2Ti2O7 are antiferromagneticly coupled [6], and the Ho3+ ions have an eﬀective ferromagnetic coupling [4]. The rare earth ions have relatively large dipoles due to unpaired localized f electrons and relatively small overlap of wavefunctions, so the dipole interaction energy is of the same magnitude as the exchange energy, unlike in transition metals where the dipole-dipole interaction is negligible. Both ions have moments of about

2− 3+ 10µB , but coordination with the O ions surrounding the Tb ions in Tb2Ti2O7 reduces its permanent moment to just over 5µB [4, 7]. Both have highly anisotropic crystal ﬁelds which force the spins to point either directly into or out of the tetrahedra in the h111i direction, making them eﬀectively Ising spins. This has been shown from a combination of magnetic susceptibility mea-

6 surements, inelastic neutron-scattering measurements, heat capacity, and theoretical crystal ﬁeld calculations [4, 7]. The single ion ground states are doublets, with the

ﬁrst excited state about 220 K above for Ho2Ti2O7 [8] and only 15-20 K above for

Tb2Ti2O7 [7].

The ground state of Tb2Ti2O7 appears to be a spin liquid. In this state, which is also called a cooperative paramagnet, there is no transition to long range magnetic order despite the development of strong short range interactions. This ground state was theoretically predicted for a pyrochlore lattice, but Tb2Ti2O7 was the ﬁrst compound to show promise of staying completely disordered all the way to zero Kelvin.

The spins remain randomly oriented and dynamic down to the lowest temperatures measured, which is at least 50 mK, even though AF correlations develop below ≈ 50 K and θ ≈ -18 K [6].

The ground state of Ho2Ti2O7 is a spin ice. It has an arrangement analogous

to the proton ordering in water ice, with two spins on each tetrahedron pointing in, and two out, which leads to the largest possible moment on every local tetrahedron, shown in Figure 4. This is because, despite susceptibility measurements indicating AF interactions, the strong dipole interaction leads to an overall ferromagnetic inter-

action. This large zero temperature degeneracy is equivalent to the frozen in residual entropy in water ice discovered by Pauling [4]. While the residual entropy is diﬃcult to recover in water ice, the application of a magnetic ﬁeld was used successfully to release the entropy in spin-ice for Ho2Ti2O7

[4] and for Dy2Ti2O7 [15]. In other words, a magnetic ﬁeld can be used to destroy the spin-ice state by breaking the large degeneracy that arises from the spin ice rules (2 spins in, 2 spin out of each tetrahedron). In this thesis I will study the way in which the frustration is relieved in magnet- ically frustrated pyrochlore systems. This will include both spin liquid and spin ice states that are disturbed by chemical substitution and/or applied magnetic ﬁelds. By

7 its nature, geometric frustration is highly dependent on the physical structure of the crystal lattice. Substitutions of diﬀerent atoms at random sites in the lattice would be expected to break the symmetries that the degenerate states rely on and relieve the

frustration. Replacing some of the titanium sites in Tb2Ti2O7 with larger tin atoms distorts the lattice, altering the positions of the magnetic terbium ions. Replacing some of the holmium atoms in Ho2Ti2O7 with chemically similar, but non-magnetic, lanthanum and yttrium atoms will create holes in the magnetic lattice, eﬀectively iso- lating some individual spins from neighbors. Various amounts of these dopants were substituted into these compounds. Subsequent, heat capacity and ac magnetization measurements were performed at various applied magnetic ﬁeld strengths and temperatures in order to measure the strength of the frustration in the altered lattices, and to investigate the nature of the frustration by testing the conditions under which the materials have frustration relieved in the form of long range ordering.

8 CHAPTER 2

EXPERIMENTAL PROCEDURE

Polycrystalline samples of Tb2Ti2−xSnxO7 for x = 0, 0.05, 0.1, 1, and 2, as well as Ho2−xRxTi2O7 for R= La with x = 0.1, 0.4, 1.7 and R= Y with x= 1.3, 1.5, and 1.8, were all obtained from an outside source. They were made by ﬁring stoichio- metric amounts of the appropriate oxides at high temperatures for several days with intermittent grindings as described elsewhere [17]. All measurements were performed in a Quantum Design PPMS, shown in Figure 5. Using various optional parts, the system can be conﬁgured to perform heat capacity, magnetization, and resistivity measurements in a temperature range of 0.36 K to 400

K and in magnetic ﬁelds up to 9 T. Temperature is maintained through a balance of resistive heating and evaporative cooling of liquid helium. Liquid 4He boils at 4.18 K at atmospheric pressure, but by pumping on it to reduce pressure the boiling point is lowered and the system can reach 1.8 K. A separate self-contained 3He system, shown in Figure 6, can be used to reach temperatures down to 0.36 K. The boiling point of 3He is 3.2 K, so the 4He system is used to ﬁrst liquefy the 3He, and then it is pumped on in order to reach lower temperatures. 3He is much rarer than 4He, so the 3He system is sealed so that the gas can be continuously reused.

A magnetic ﬁeld is provided by a superconducting solenoid around the sample chamber, inside the same helium reservoir used for the 4He cooling. A persistent current up to 46.7 A is sustained in the solenoid to generate the large magnetic ﬁelds.

Heat Capacity Measurement

The heat capacity of the samples is measured using a quasi-adiabatic thermal relaxation technique. The samples are ground and pressed into pellets. They are

9 then secured to an alumina stage with Apiezon N grease in order to assure good thermal contact. The stage is supported by platinum wires connected to a resistive heater and a thermometer, all inside a removable copper puck, shown in Figure 7. In order to ensure that the wires are the only signiﬁcant thermal link to the stage, the

sample chamber is evacuated to a pressure of about 0.01 mTorr using a turbo-pump. The sample puck is brought to thermal equilibrium at the desired temperature, and then a heat pulse is sent along the wires to the resistive heater on the underside of the stage. The temperature is continuously measured as the stage cools back to base temperature. The temperature rise is kept small, at about 2% of the base

temperature, in order to resolve sharp peaks in the data and to make sure that the heat capacity does not vary signiﬁcantly over the measurement. The change in temperature with time is ﬁt to a simple exponential model given by:

dT Ctotal = −KW (T − To)+ P (t) (2.1) dt

where Ctotal is the total heat capacity of the sample and platform, KW is the thermal conductance of the supporting wires, To is the base temperature of the puck, and P is the heater power. The solution is exponential with characteristic time-constant τ equal to Ctotal/K. This assumes that the coupling between the stage and sample is perfect, but if it is not close enough then a model which ﬁts to two time constants is needed: dT C p = −K (T − T )+ K (T − T ) (2.2) platform dt W p o g s

dT C s = −K (T − T ) (2.3) sample dt g s p

where Kg is the thermal conductance between the sample and platform. The heat capacity of the stage and grease alone is measured beforehand, so it can be subtracted from the total found from the ﬁt to ﬁnd the sample heat capacity. The simpler model

10 is used when measuring with no sample or for a very small sample, because the two-tau model diverges for 100% sample coupling. This technique measures heat capacity at constant pressure, not constant volume. The change in volume of solids with pressure and temperature is small (at least for the temperatures measured in the current study), so Cp and Cv are considered equivalent in our results.

The heat capacities of all Tb2Ti2−xSnxO7 samples were measured to the lowest temperature possible, about 0.4 K, and in magnetic ﬁelds from 0 to 9 Tesla.

Magnetic Mesurements

The PPMS functions as both a DC magnetometer and as an AC susceptometer. A set of copper drive and detection coils is inserted within the superconducting magnet, shown in Figure 8. The sample is moved through the coils using a servo motor and

the induced signal in the detection coils is analyzed. The detection coils are arranged in a ﬁrst order gradiometer conﬁguration with two sets of counter-wound copper wires connected in series separated by several centimeters. A larger AC drive coil surrounds them both, which is also surrounded by and connected in series with a counter-wound

compensation coil that reduces interaction with materials outside the coils. The AC Magnetization option for the PPMS is not compatible with the 3He option, so the minimum temperature achievable is 1.8 K. During DC measurements the sample is put in a constant persistent ﬁeld, H, using the superconducting magnet, creating a moment, M, in the sample. It is then moved quickly through the detection coils and a signal is induced according to Faraday’s Law. The magnetic susceptibility is calculated as:

M χ = (2.4) H

For AC measurements, a small alternating ﬁeld, dH, is applied to the sample chamber

11 using the drive coil, while a large constant ﬁeld can still be applied by the superconducting magnet. The amplitude and phase of the sample’s response is received by the detection coils and compared with the drive signal. The change in the samples moment, dM, is found from this. The AC susceptibility is then calculated by:

dM χ = (2.5) AC dH

This is the local slope of the sample’s magnetization curve. The ideal sample response signal is 90◦ out of phase with the drive signal in accordance with Faraday’s law, but other factors lead to deviations. Therefore, the AC susceptibility can also be described by a real component, χ0, that is in phase with the ideal response and an imaginary component, χ00, that is 90◦ out of phase with the ideal response. These

values are indicative of the “stiffness” of the spins, or their resistance to flipping with the changing field.

The DC magnetizations of all the Ho2−xRxTi2O7 samples were measured in a 0.1 Tesla persistent field down to the lowest temperature. Their AC susceptibilities were then measured in a 1 Tesla persistent field with a 10 Oe oscillating field at frequencies from 10 Hz to 10 kHz at temperatures from about 100 K down to 1.8 K as well. When spins ‘slow down’ (spin relaxation times can be as fast 10−14 s), our ac susceptibility measurements allow the exploration of time scales from 10−1 to

10−4 s. When combined with collaborators who perform neutron scattering, we can span time scales to 10−14 s allowing a wide dynamic measurement range.

12 CHAPTER 3

RESULTS AND ANALYSIS

Sn substituion in Tb2Ti2O7

The magnetic heat capacities of all the Tb2Ti2−xSnxO7 samples from about 3 K to the lowest measured temperatures and at various applied field strengths are shown in Figures 9 through 13. Non-magnetic Y2Ti2O7 was used to estimate the lattice contribution to the total heat capacity, and it was found that the lattice contribution was very small compared to the dominating magnetic contribution at these temperatures. The errors in the heat capacity measurements are on the order of 0.1%, which is smaller than the data points in the graphs. In Figure 9, the zero applied field graph shows the spin liquid behavior of the pure Tb2Ti2O7 sample down to our lowest applied temperature, as no features corresponding to coordination between the spins or a sharp feature indicating a transition to long range order are seen. All the other graphs show an upturn below 1 K that increases with applied field, which is believed to be part of a nuclear Schottky peak.

This is caused by static Terbium electron spins reacting to the applied field, creating a very large internal field to which the Tb nuclear moments react. This shows that many Tb spins are static as they line up with each other, which means that the applied field breaks the spin liquid state [18]. A sharper feature at about 0.8 Kelvin in

Figure 13 indicates that the pure Tb2Sn2O7 transitions to what may be a type of a long range ordered spin ice state in zero field, even though there is evidence of slow magnetic fluctuations persisting that are similar to the spin liquid state [19]. The strong internal field splits the degenerate ground state by an energy gap .

13 The heat capacity for such a two level magnetic system is given by:

2 ∆E ∆E e( kT ) C = Nk 2 (3.1) kT ∆E 1+ e( kT )

In the high temperature limit, the trailing edge of the Schottky peak can be approximated by: A C ≈ (3.2) T 2 where, for nuclear spins:

2 Nk I +1 µ B A = N hyp (3.3) 3 I k

with I as the nuclear spin of the magnetic atom, which is 3/2 for our Terbium, µN

is a nuclear magneton, and Bhyp equal to the hyperﬁne magnetic ﬁeld at the site of the magnetic atom. If there is a low temperature upturn in our heat capacity data, we can attribute it to the appearance of this nuclear entropy. It is important to note that this will only occur when the nuclear moments experience large static (at least on the order of heat capacity measurements which are typically many seconds)

magnetic ﬁelds from the electron magnetic moments. This gives us a thermodynamic measurement of static magnetic moments that are typically involved in short or long- range magnetic order that can be determined by neutron scattering measurements. As our measurements are relatively easy to perform, they are a very useful ‘ﬁrst step’

measurement to lead the way in performing more time and infrastructure intensive nuclear scattering measurements.

A, and therefore Bhyp, can be found by a least squares ﬁt to the data below 1 K to: A C = + b T 2 + c T (3.4) T 2

where the T 2 and T terms account for the non-nuclear (namely magnetic and lattice

14 as the systems in question are insulators) contributions. The hyperfine field was calculated from A, using equation (3.3). The results of these fits and calculations are collected in Tables 2 and 3, and shown in Figures 14 through 18. Errors are taken from the least squared fits from equation (3.4). In general, all the doped samples

showed an increase in the hyperfine field up to about a 2 T applied field, where it then seemed to almost level off at just below 20 T, and then increased very slowly

from there. The pure Tb2Ti2O7 sample showed a fairly steady increase to about 22.5

T without leveling off. The Tb2Sn2O7 sample showed what could be two plateaus; one from 2 to 3 T applied fields and another from 6 T on, although their significance is unknown.

La and Y substitution in Ho2Ti2O7

The graphs of the imaginary part of the AC susceptibility divided by the DC magnetization versus temperature for each applied oscillating frequency show peaks which correspond to the temperature for which the applied frequency becomes larger than the actual spin frequency. These peaks occur because at lower frequencies the spins can more easily keep up with the changing magnetic field, and so will be more in phase with the drive field, going to just the DC response in the extreme low frequency limit. In the opposing limit, a frequency that is too high will result in the spins not having enough time to come into alignment with the oscillating field, reducing the magnitude of the in phase AC response.

In order for a single spin in one of the sample’s two-in-two-out magnetic tetrahedra to ﬂip, it must overcome the energy barrier between the degenerate ground state and the ﬁrst-excited state. This is expected to result in Arrhenius, or thermally activated, behavior which is described by the equation:

∆ Tpeak t = toe (3.5)

15 where to is the characteristic time between spin ﬂips of the sample, t is the inverse of

the AC drive coil frequency f, the peak temperature at that frequency is Tpeak, and ∆ is the energy barrier between states in Kelvin. Rearranging equation (3.5) gives:

1 −1 1 = ln f + ln to (3.6) Tpeak ∆ ∆

From this equation, one would expect that a graph of inverse peak temperature versus

the natural log of the input frequency would give a straight line whose slope could be used to determine the energy diﬀerence between the ground and ﬁrst excited states, as in Figure 19. An example of the utility of AC susceptibility measurements can be seen in past

measurements of the spin ice compound Dy2Ti2O7 [20]. These results in Figure 20

clearly show multiple relaxation processes. The ‘high temperature’ process at about 15 K has an energy barrier of about 191 K and is due to single spin flips being promoted to the first excited crystal field level. The low temperature process at about 2 K only has an enrgy barrier of about 8.9 K and is believed to be due to a

cooperative eﬀect involving multiple spin ﬂips.

The results of the AC susceptibility measurements for the spin-ice Ho2−xRxTi2O7 samples are shown in Figures 21 through 26.

All the χAC /χDC graphs show a larger peak at from 10 K to 20 K, and a smaller peak below 5 K, though the smaller peak could not be distinguished well due to either becoming lost in the tail of the larger peak or quickly shifting out of our temperature range. The Y doped samples showed that the larger peak increases in intensity and shifts to steadily higher temperatures with increasing drive ﬁeld frequency, while the La doped samples showed a very diﬀerent trend, even for samples that had similar

ratios of Ho to non-magnetic atoms. All three La samples showed an initial decrease in peak intensities with rising frequencies, while the peak did not seem to shift at all.

16 Then the two samples that were least highly doped with x =0.4 and x = 0.1 showed a very abrupt shift to higher temperature and reversed direction to start increasing with rising frequencies. The larger peaks were fit to a Gaussian function in order to find their centers. The graphs of the peak temperatures versus the logarithm of the applied field frequency are shown in Figure 27. All errors are taken from the least squares fits to the Gaus- sian function. None of these graphs entirely show the Arrhenius behavior we were expecting. A value of about 270 K for the energy barrier was found in reference [8] using neutron spin echo (NSE) data which probes much higher frequencies ( 107 to

1011 Hz), but no parts of the data we obtained could be fit to obtain a similar value for ∆. Instead, large ranges of the frequencies showed no clear dependence of the peak temperatures on the resonant frequencies. The yttrium samples show plateaus of unchanging behavior up to about 120 Hz, then what looks like a linear trend that appears to be the expected Arrhenius behavior for the rest of the higher frequency range. Fits to the data for Y = 1.3, 1.5, and 1.8 showed values for ∆ of 98.8 K ± 3.5 K, 94.5 K ± 3.6 K, and 92.17 K ± 6.0 K respectively. This difference from the NSE data may be due to the presence of the other process with a similar time scale which caused the smaller peak. There appears to be no evidence of thermally activated behavior in the lanthanum samples for the entire range of frequencies. In the La = 1.7 sample, the peak temperature remains relatively constant, but the La = 0.4 and 0.1 samples show a sudden jump to another temperature at about 500 Hz, which then remains constant again. This difference in the highly doped La samples is because the larger La ions exert more chemical pressure on the lattice, forming a coexisting a non- pyrochlore phase for x > 0.4 [8]. The regions where the peaks become independent of temperature may be due to slower quantum relaxation processes, which essentially need zero activation energy, which become dominant at lower temperatures where the thermal process freezes.

17 CHAPTER 4

CONCLUSIONS

The most surprising results of the heat capacity measurements of the Tb2Ti2−xSnxO7 samples were that, despite the large amount of structural disorder introduced, none of the doped samples showed any abrupt transitions to a long range magnetic ordered state at any temperatures or fields in our measurement ranges. The upturns in the data at low temperatures for the applied fields do confirm that the spin liquid state is broken, though. In order to produce these upturns, at least some of the Tb spins must be static over the time scale of the measurements, which are on the order of seconds. This means that long range order does occur at low temperatures, but it develops slowly from short range order instead of from a collective phase transition.

It is also noteworthy that an applied ﬁeld of 2T makes the pure Tb2Ti2O7 behave very similarly to Tb2Sn2O7 [21]. The magnetic lattice likely transitions to a spin ice state like that of Ho2Ti2O7 with correlations over single tetrahedrons, but still no translational symmetry over the entire lattice.

Though the results from the Ho2−xRxTi2O7 samples do not necessarily match earlier NSE measurements, the differences can be explained by processes that are not active at the much higher frequencies that those techniques must be used. Further work would need to be done in order to better understand the exact nature of the new processes that these results reveal. The results from these different samples are very similar, despite very large substitutions of holes in the magnetic lattices which effectively isolate individual spins from each other. The frustrated spin-spin interactions are very robust, with little change in the crystal field and spin dynamics, especially in the Y doped samples. While single crystal neutron measurements may be better for directly measuring the dynamics of individual spins in the lattice, bulk measurements like heat capacity

18 and AC susceptibility which can indirectly probe similar properties are much easier, faster, and cost eﬀective. This makes them excellent compliments to each other, so that areas of likely interest can be identiﬁed using easier techniques, which can then be more thoroughly analyzed at a neutron source.

19 Figure 1 A triangular lattice with antiferromagnetically coupled Ising spins at its corners. The interaction between the two spins at the top and left are satisﬁed, but the third cannot be satisﬁed with both of the others at the same time, making neither state energetically favorable.

Figure 2 Taken from Ref. [10]. The graph on the left shows the expected behavior of a nonfrustrated antiferromagnetic material, which diverges from linear Curie-Weiss behavior near its ordering temperature, TN , which is about equal to its projected Curie-Weiss Tempurature, θCW . The graph on the left represents a strongly frustrated material with the same θCW , which stays linear until it orders at a much lower temperature.

20 Figure 3 Taken from Ref. [16]. The pyrochlore lattice is an arrangement of corner sharing tetrahedra. In Tb2Ti2O7 and Ho2Ti2O7, magnetic atoms sit on each corner and their spins are conﬁned to point directly into or out of each tetrahedron.

Figure 4 Taken from Ref. [4]. (A) shows the proton ordering in water ice, with two hydrogen atoms nearer each oxygen and two farther. (B) shows a single tetrahedron from the Ho2Ti2O7 lattice showing the same arrangement, but with the hydrogen atoms replaced by vectors at the corners representing the direction of the spins.

21 22

Figure 5 A Quantum Design Physical Property Measurement System, with an interior view of the dewar. 23

Figure 6 The Helium 3 insert, which allows the system to reach 0.36 Kelvin by pumping on a self enclosed liquid helium reservoir. 24

Figure 7 A Helium 3 Heat Capacity Puck. The left image shows the placement of the heater and the thermometer on the underside of the stage on which the sample is placed. 25

Figure 8 The drive and detection coils for the PPMS ACMS Option. They reside within the larger superconducting magnet. Tb Ti O

2 2 7

0 kOe

4 20 kOe

40 kOe

60 kOe (J/mol-K) p 80 kOe

C 2

0 1 2 3

T (K)

Figure 9 The magnetic heat capacity of the undoped terbium titanate for several applied fields. The zero field measurement shows an absence of correlated behavior for its spins, which is broken with the introduction of an applied field. No sharp transition to long range order is seen at any field. Errors are on the order of 0.1%, which is smaller than the data points.

26 Tb Ti Sn O

2 1.95 0.05 7

0 kOe

5 kOe

10 kOe

20 kOe

40 kOe

60 kOe (J/mol-K) p 2 80 kOe C

0 1 2 3

T (K)

Figure 10 The magnetic heat capacity of Tb2Ti1.95Sn0.05O7 at several applied ﬁelds. No sharp transition to long range order is seen.

27 Tb Ti Sn O

2 1.9 0.1 7

0 kOe

2 kOe

5 kOe

10 kOe

15 kOe

20 kOe

30 kOe

40 kOe (J/mol-K) p

50 kOe C

60 kOe

70 kOe

0 1 2 3

T (K)

Figure 11 The magnetic heat capacity of Tb2Ti1.9Sn0.1O7 at several applied ﬁelds. No sharp transition to long range order is seen.

28 Tb TiSnO

2 7

0 kOe

5 kOe

10 kOe

15 kOe

20 kOe

40 kOe

60 kOe (J/mol-K) p

80 kOe C

0 1 2 3

T (K)

Figure 12 The magnetic heat capacity of Tb2TiSnO7 at several applied ﬁelds. No sharp transition to long range order is seen.

29 Tb Sn O

2 2 7

0 kOe

5 kOe

10 kOe (J/mol-K)

15 kOe p

2 C

20 kOe

0 1 2 3

T (K)

Figure 13 The magnetic heat capacity of Tb2Sn2O7 at several applied ﬁelds. No sharp transition to long range order is seen.

30 Tb Ti O

2 2 7

300

250

200

150

100 Hyperfine Field (kOe)

0 20 40 60 80

Applied field (kOe)

Figure 14 The hyperfine field at the terbium sites for the pure Tb2Ti2O7 as it varies with the applied field.

31 Tb Ti Sn O

2 1.95 0.05 7

220

200

180

160

140

120

100

60 Hyperfine Field (kOe)

0 10 20 30 40 50 60 70 80 90

Applied Field (kOe)

Figure 15 The hyperfine field at the terbium sites for Tb2Ti1.95Sn0.05O7 as it varies with the applied field.

32 Tb Ti Sn O

2 1.9 0.1 7

220

200

180

160

140

120

100

60 Hyperfine Field (kOe)

-10 0 10 20 30 40 50 60 70 80

Applied Field (kOe)

Figure 16 The hyperfine field at the terbium sites for Tb2Ti1.95Sn0.05O7 as it varies with the applied field.

33 Tb TiSnO

2 7

220

200

180

160

140

120

100

60 Hyperfine Field (kOe)

0 10 20 30 40 50

Applied Feild (kOe)

Figure 17 The hyperfine field at the terbium sites for Tb2Ti1.95Sn0.05O7 as it varies with the applied field.

34 Tb Sn O

2 2 7

200

180

160

140

120

100

60 Hyperfine Feild (kOe)

0 10 20 30 40 50 60 70 80

Applied Feild (kOe)

Figure 18 The hyperfine field at the terbium sites for Tb2Sn2O7 as it varies with the applied field.

35 Figure 19 The expected response due to thermally activated behavior. The graph of the logarithm of applied frequency versus the inverse of the peak temperature should conform to a straight line. When ﬁt to equation 3.6, the slope of the line should equal the inverse of the energy barrier, and the y-intercept can be used to calculate the characteristic time between the spins.

36 37

Figure 20 Taken from Ref. [20]. (left) (a) The normalized imaginary AC susceptibility versus temperature for Dy2Ti2O7 for different applied DC fields. (b) The solid lines show peaks from two separate processes at about 2 K and 15 K. A third peak exists between them, but it is difficult the separate. (right) The inverse of the maximum peak temperature versus the log f. The linear fits give values of 191K and 8.9K for the high and low temperature peaks, respectively. Ho Y Ti O

0.2 1.8 2 7

10 Hz

0.3

464 Hz

1000 Hz

2154 Hz

4641 Hz

7600 Hz

0.2 DC

10000Hz

X''/ X

0.1

0.0

0 10 20 30 40

Temperature (K)

Figure 21 The AC susceptibility of Ho0.2Y1.8Ti2O7 divided by its DC magnetization for some of the frequencies that were investigated from 10 Hz to 10 kHz. There appears to be a smaller peak at about 5 Kelvin, but it is lost in the larger one. The large peak grows in magnitude and shifts left with higher frequency.

38 Ho Y Ti O

0.5 1.5 2 7

10 Hz

464 Hz

0.27 1000 Hz

2154 Hz

4641 Hz

7600 Hz

10000 Hz DC 0.18

X'' / X

0.09

0.00

0 10 20 30 40

Temperature (K)

Figure 22 The AC susceptibility of Ho0.5Y1.5Ti2O7 divided by its DC magnetization for some of the frequencies that were investigated from 10 Hz to 10 kHz. There appears to be a smaller peak at about 5 Kelvin, but it is lost in the larger one. The large peak grows in magnitude and shifts left with higher frequency.

39 Ho Y Ti O

0.7 1.3 2 7

10 Hz

464 Hz

1000 Hz

0.2 2154 Hz

4642 Hz

7600 Hz

10000 Hz DC

0.1 X'' / X

0.0

0 10 20 30 40

Temperature (K)

Figure 23 The AC susceptibility of Ho0.7Y1.3Ti2O7 divided by its DC magnetization for some of the frequencies that were investigated from 10 Hz to 10 kHz. There appears to be a smaller peak at about 5 Kelvin, but it is lost in the larger one. The large peak grows in magnitude and shifts left with higher frequency.

40 Ho La Ti O

0.3 1.7 2 7

10 Hz

24.54 Hz

75 Hz

0.01 215.44 Hz

2154 Hz DC

7600 Hz

10000 Hz X'' / X

0.00

0 10 20 30

Temperature (K)

Figure 24 The AC susceptibility of Ho0.3La1.7Ti2O7 divided by its DC magnetization for some of the frequencies that were investigated from 10 Hz to 10 kHz. A smaller peak at about 5 Kelvin is lost in the larger feature. Unlike the Y doped samples, the larger peak gets smaller with increasing frequencies, and it shows no clear trend to higher or lower temperatures.

41 Ho La Ti O

1.6 0.4 2 7

0.050

10 Hz

0.045

46.4 Hz

0.040 215 Hz

1000 Hz

0.035

2154 Hz

4642 Hz

0.030

7600 Hz DC

0.025

10000 Hz X'' / X 0.020

0.015

0.010

0.005

0.000

0 10 20 30 40

Temperature (K)

Figure 25 The AC susceptibility of Ho1.6La0.4Ti2O7 divided by its DC magnetization for some of the frequencies that were investigated from 10 Hz to 10 kHz. A smaller peak can be seen below 5 Kelvin, but it moves out of the observable range at lower frequencies. There is still only one larger peak. It seems to remain at a constant temperature of about 10 K and decreases with increasing frequency, and then moves to a higher constant temperature of about 18 K and increases again with farther increasing frequency.

42 Ho La Ti O

1.9 0.1 2 7

10 Hz

0.04

46.4 Hz

215 Hz

1000 Hz

2154 Hz

4642 Hz DC

7600 Hz

10000 Hz

0.02 X'' / X

0.00

0 10 20 30 40

Temperature (K)

Figure 26 The AC susceptibility of Ho1.9La0.1Ti2O7 divided by its DC magnetization for some of the frequencies that were investigated from 10 Hz to 10 kHz. A smaller peak can be seen below 5 Kelvin, but it moves out of the observable range at lower frequencies. There is still only one larger peak. It seems to remain at a constant temperature of about 10 K and decreases with increasing frequency, and then moves to a higher constant temperature of about 18 K and increases again with farther increasing frequency.

43 Ho La Ti O Ho Y Ti O

0.3 1.7 2 7

0.2 1.8 2 7

0.13

0.12

0.14

0.11

0.10 0.12

0.09

0.10

0.08 ) ) -1 -1

0.07

0.08 (K (K

0.06 Peak Peak

0.05 0.06 1 / T / 1 1 1 / T

0.04

0.03

0.02

0.01

0.00 0.00

100 1000 10000 100 1000 10000

Applied Frequency (Hz) Applied Frequency (Hz)

Ho La Ti O

Ho Y Ti O 1.6 0.4 2 7

0.5 1.5 2 7

0.10

0.11

0.09

0.10

0.08

0.09

0.07 0.08

0.07

0.06 ) -1 ) -1

0.06

(K 0.05

0.05 Peak

0.04 1 1 / T (K

0.04 1 1 / T

0.03

0.02

0.01

0.00 0.00

100 1000 10000 100 1000 10000

Applied Freqency (Hz) Applied Frequency (Hz)

Ho La Ti O

Ho Y Ti O 1.9 0.1 2 7

0.7 1.3 2 7

0.10

0.09

0.08

0.07

0.06 ) )

0.06 -1 -1

(K 0.05 (K Peak Peak

0.04

0.04 1/T 1 1 / T

0.03

0.02

0.01

0.00 0.00

100 1000 10000 100 1000 10000

Applied Frequency (Hz) Applied Frequency (Hz)

Figure 27 Graphs of the applied frequency on a logarithmic scale versus the inverse peak frequency for all six holmium titanate samples. None of the samples seem to conform to the expected Arrhenius behavior. All samples show regions where frequency appears entirely independent of the temperature.

44 Table 1 Frustration parameters for some strongly frustrated materials with frustration parameters f = |θ|/Tmag greater than 10, and some more conventional materials for comparison. Tb2Ti2O7 and Ho2Ti2O7 have not been observed to transition to long range order down to the lowest observed temperatures.

45 Table 2 The results of the fit of the low temperature heat capacity of Tb2Ti2−xSnxO7 for x = 0, 0.05, and 0.1 to a high temperature Schottky peak and the calculation of the hyperfine field. Bhyp was not calculated for poor fits that resulted in negative values for A. Errors are the standard deviation of the parameter.

46 Table 3 The results of the fit of the low temperature heat capacity of Tb2Ti2−xSnxO7 for x = 1 and 2 to a high temperature Schottky peak and the calculation of the hyperfine field. Bhyp was not calculated for poor fits that resulted in negative values for A. Errors are the standard deviation of the parameter.

47 REFERENCES

[1] Wannier, G. H., Antiferromagnetism - the Triangular Ising Net. Physical Review 79, 357-364 (1950). [2] Enjalran, M., Gingras, M.J.P., Kao, Y-J, Del Maestro, A., and Molavian, H. R. The spin liquid state of the Tb2Ti2O7 pyrochlore antiferromagnet: a puzzling state of aﬀairs. Journal of Physics: Condensed Matter 16, S673-S678 (2004).

[3] Rule, K.C., Ruﬀ, J.P.C., Gaulin, B.D., et al. Field-induced order and spin waves in the pyrochlore antiferromagnet Tb2Ti2O7. Physical Review Leters 96 177201 (2006). [4] Bramwell, S.T. and Gingras, M.J.P. Spin ice state in frustrated magnetic pyrochlore materials. Science 294, 1495-1501 (2001). [5] Diep, H.T. Magnetic Systems with Competeing Interactions. World Sci., (1994). [6] Gardner, J.S., Keren, A., Ehlers, G., et al. Dynamic fraustrated magnetism in Tb2Ti2O2 at 50 mK. Physical Review B 68, 180401 (2003).

[7] Gingras, MJP, den Hertog, BC, Faucher, M, et al. Thermodynamic and single- ion properties of Tb3+ within the collective paramagnetic-spin liquid state of the frustrated pyrochlore antiferromagnet Tb2Ti2O7. Physical Reveiw B 62 6496-6511 (2000). [8] Ehlers, G; Gardner, JS; Booth, CH, et al. Dynamics of diluted Ho spin ice Ho2−xYxTi2O7 studied by neutron spin echo spectroscopy and ac susceptibility. Physical Review B 73 174429 (2006). [9] Kittel, Charles Introduction to Solid State Physics. Hoboken, NJ: John Wiley Sons, Inc. (2005).

[10] Ramirez, AP Strongly Geometrically Frustrated Magnets. Annual Review Of Ma- terials Science 24 453-480 (1994). [11] Hubbard, J MAGNETISM OF IRON .2. Physical Review B 20 4584-4595 (1979). [12] Turek, I; Blugel, S; Bihlmayer, G, et al. Exchange interactions at surfaces of Fe, Co, and Gd . CZECHOSLOVAK JOURNAL OF PHYSICS 53 81-88 (2003). [13] Hall, H. E. Solid State Physics. John Wiley Sons, Ltd. (1974) [14] Gottfried, K. and Yan Tung-Mow Quantum Mechanics: Fundamentals. Springer- Verlag, New York, LLC (2004)

[15] Ramirez, AP; Hayashi, A; Cava, RJ, et al. Zero-point entropy in ‘spin ice’. Nature 399 333-335 (1999). [16] Moessner, R. Magnets with strong geometric frustration. Canadian Journal of Physics 79 1283-1294 (2001).

48 [17] Han, SW; Gardner, JS; Booth, CH Structural properties of the geometrically frustrated pyrochlore Tb2Ti2O7. Physical Review B 69 024416 (2004). [18] Cornelius, AL; Light, BE; Kumar, RS, et al. Disturbing the spin liquid state in Tb2Sn2O7: Heat capacity measurements on rare earth titanates. Physica B 359 1243-1245 (2005). [19] Mirebeau, I; Apetrei, A; Rodriguez-Carvajal, J, et al. Ordered spin ice state and magnetic ﬂuctuations in Tb2Sn2O7. Physical Reveiw Letters 94 157205 (2005). [20] Cornelius, A.L., Yulga, B.F., Light, B.E., Gardner, J.S. Observation of Multiple Spin Relaxation Processes in the Spin-Ice Compound Dy2Ti2O7. Unpublished (2002) [21] Rule, K. C.,Ehlers, G.,Stewart, J. R.,Cornelius, A. L.,Deen, P. P.,Qiu, Y.,Wiebe, C. R.,Janik, J. A.,Zhou, H. D.,Antonio, D.,Woytko, B. W.,Ruﬀ, J. P.,Dabkowska, H. A.,Gaulin, B. D.,Gardner, J. S. Polarized inelastic neutron scattering of the partially ordered Tb2Sn2O7. Pysical Review B, 76 212405 (2007)

49 VITA

Graduate College University of Nevada, Las Vegas

Daniel Antonio

Degrees: Bachelors of Science, Physics, 2004 University of Nevada, Las Vegas

Publications: Christianson A.D., Lawrence J.M., Zarestky J.L., Suzuki H.S., Thompson J.D., Hundley M.F., Sarrao J.L., Booth C.H., Antonio D., Cornelius A.L. Antiferromagnetism in Pr3In. Physical Review B 72 024402 (2005)

Ehlers, G., Gardner, J. S., Booth, C. H., Daniel, M., Kam, K. C., Cheetham, A. K., Antonio, D., Brooks, H. E., Cornelius, A. L., Bramwell, S. T., Lago, J., Haeussler, W., Rosov, N. Dynamics of diluted Ho spin ice Ho2−xYxTi2O7 studied by neutron spin echo spectroscopy and ac susceptibility. Physical Review B 73 174429 (2006)

Rule, K. C.,Ehlers, G.,Stewart, J. R.,Cornelius, A. L.,Deen, P. P.,Qiu, Y.,Wiebe, C. R.,Janik, J. A.,Zhou, H. D.,Antonio, D.,Woytko, B. W.,Ruﬀ, J. P.,Dabkowska, H. A.,Gaulin, B. D.,Gardner, J. S. Polarized inelastic neutron scattering of the partially ordered Tb2Sn2O7. Pysical Review B, 76 212405 (2007).

Thesis Title: AC Susceptibility and Heat Capacity Studies of the Geometrically Frustrated Pyrochlores Tb2Ti2−xSnxO7 and Ho2−xRxTi2O7

Thesis Committee: Committee Chairperson: Dr. Andrew Cornelius, Ph.D. Committee Memeber: Dr. Micheal Pravica, Ph.D. Committee Memeber: Dr. Tao Pang, Ph.D. Graduate Faculty Representative: Dr. Adam Simon, Ph.D.